Integrand size = 23, antiderivative size = 182 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=-\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \] Output:
-2*a/d/e/(e*csc(d*x+c))^(1/2)-2/3*a*cos(d*x+c)/d/e/(e*csc(d*x+c))^(1/2)+a* arctan(sin(d*x+c)^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+a*arcta nh(sin(d*x+c)^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+2/3*a*Inver seJacobiAM(1/2*c-1/4*Pi+1/2*d*x,2^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+ c)^(1/2)
Time = 9.48 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.74 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=-\frac {a \left (12+4 \cos (c+d x)+6 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\csc (c+d x)}+3 \sqrt {\csc (c+d x)} \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \sqrt {\csc (c+d x)} \log \left (1+\sqrt {\csc (c+d x)}\right )+\frac {4 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )}{\sqrt {\sin (c+d x)}}\right )}{6 d e \sqrt {e \csc (c+d x)}} \] Input:
Integrate[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(3/2),x]
Output:
-1/6*(a*(12 + 4*Cos[c + d*x] + 6*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Csc[c + d *x]] + 3*Sqrt[Csc[c + d*x]]*Log[1 - Sqrt[Csc[c + d*x]]] - 3*Sqrt[Csc[c + d *x]]*Log[1 + Sqrt[Csc[c + d*x]]] + (4*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]) /Sqrt[Sin[c + d*x]]))/(d*e*Sqrt[e*Csc[c + d*x]])
Time = 0.66 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.70, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 3317, 3042, 3044, 262, 266, 756, 216, 219, 3115, 3042, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a \sec (c+d x)+a}{(e \csc (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4366 |
\(\displaystyle \frac {\int (\sec (c+d x) a+a) \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \frac {\int -\left ((-\cos (c+d x) a-a) \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)\right )dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -\left ((\cos (c+d x) a+a) \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)\right )dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int (\cos (c+d x) a+a) \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\left (-\cos \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3317 |
\(\displaystyle \frac {a \int \sin ^{\frac {3}{2}}(c+d x)dx+a \int \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \int \sin (c+d x)^{3/2}dx+a \int \frac {\sin (c+d x)^{3/2}}{\cos (c+d x)}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {\frac {a \int \frac {\sin ^{\frac {3}{2}}(c+d x)}{1-\sin ^2(c+d x)}d\sin (c+d x)}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {a \left (\int \frac {1}{\sqrt {\sin (c+d x)} \left (1-\sin ^2(c+d x)\right )}d\sin (c+d x)-2 \sqrt {\sin (c+d x)}\right )}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {a \left (2 \int \frac {1}{1-\sin ^2(c+d x)}d\sqrt {\sin (c+d x)}-2 \sqrt {\sin (c+d x)}\right )}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {a \int \sin (c+d x)^{3/2}dx+\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}+\frac {1}{2} \int \frac {1}{\sin (c+d x)+1}d\sqrt {\sin (c+d x)}\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}+\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a \int \sin (c+d x)^{3/2}dx+\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {a \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )+\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )+\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}+a \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
Input:
Int[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(3/2),x]
Output:
((a*(2*(ArcTan[Sqrt[Sin[c + d*x]]]/2 + ArcTanh[Sqrt[Sin[c + d*x]]]/2) - 2* Sqrt[Sin[c + d*x]]))/d + a*((2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d) - ( 2*Cos[c + d*x]*Sqrt[Sin[c + d*x]])/(3*d)))/(e*Sqrt[e*Csc[c + d*x]]*Sqrt[Si n[c + d*x]])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos [e + f*x]^FracPart[p] Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / ; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 1.84 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.76
method | result | size |
parts | \(\frac {a \sqrt {2}\, \left (i \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )-\sqrt {2}\, \cos \left (d x +c \right )\right )}{3 d e \sqrt {e \csc \left (d x +c \right )}}+\frac {a \left (-2 \cos \left (d x +c \right )+2\right ) \left (\arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sin \left (d x +c \right )-\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \sin \left (d x +c \right )+2 \cos \left (d x +c \right )-2\right )}{2 d e \sqrt {e \csc \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right )^{2}}\) | \(320\) |
default | \(-\frac {a \sqrt {2}\, \left (\sin \left (d x +c \right ) \left (2 \cos \left (d x +c \right )+6\right ) \sqrt {2}+i \left (-3 \cos \left (d x +c \right )-3\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+i \left (-3 \cos \left (d x +c \right )-3\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+i \left (4 \cos \left (d x +c \right )+4\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\left (-3 \cos \left (d x +c \right )-3\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\left (3 \cos \left (d x +c \right )+3\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\right ) \csc \left (d x +c \right )}{6 d e \sqrt {e \csc \left (d x +c \right )}}\) | \(583\) |
Input:
int((a+a*sec(d*x+c))/(e*csc(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3*a/d*2^(1/2)/e/(e*csc(d*x+c))^(1/2)*(I*(1+I*cot(d*x+c)-I*csc(d*x+c))^(1 /2)*(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2 )*EllipticF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))*(csc(d*x+c)+c ot(d*x+c))-2^(1/2)*cos(d*x+c))+1/2*a/d*(-2*cos(d*x+c)+2)*(arctan(sin(d*x+c )*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c)))*(sin(d*x+c)/(1+cos( d*x+c))^2)^(1/2)*sin(d*x+c)-arctanh(sin(d*x+c)*(sin(d*x+c)/(1+cos(d*x+c))^ 2)^(1/2)/(-1+cos(d*x+c)))*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*sin(d*x+c)+2 *cos(d*x+c)-2)/e/(e*csc(d*x+c))^(1/2)/(-1+cos(d*x+c))^2
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.34 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))^(3/2),x, algorithm="fricas")
Output:
[-1/24*(6*a*sqrt(-e)*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)*sqr t(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e)) + 3*a*sqrt(-e)*log((e*cos (d*x + c)^4 - 72*e*cos(d*x + c)^2 - 8*(cos(d*x + c)^4 - 9*cos(d*x + c)^2 + (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)*sqrt(-e)*sqrt(e/sin(d*x + c)) + 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d *x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)) + 16*(a*cos(d*x + c) + 3*a)*sqrt(e/sin(d*x + c))*sin(d*x + c) + 8*I*a*sqrt(2*I*e)*weierstrassP Inverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 8*I*a*sqrt(-2*I*e)*weierstr assPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)))/(d*e^2), -1/24*(6*a*sqrt (e)*arctan(1/4*(cos(d*x + c)^2 + 6*sin(d*x + c) - 2)*sqrt(e)*sqrt(e/sin(d* x + c))/(e*sin(d*x + c) - e)) - 3*a*sqrt(e)*log((e*cos(d*x + c)^4 - 72*e*c os(d*x + c)^2 + 8*(cos(d*x + c)^4 - 9*cos(d*x + c)^2 - (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)*sqrt(e)*sqrt(e/sin(d*x + c)) - 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 + 4*(cos(d* x + c)^2 - 2)*sin(d*x + c) + 8)) + 16*(a*cos(d*x + c) + 3*a)*sqrt(e/sin(d* x + c))*sin(d*x + c) + 8*I*a*sqrt(2*I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 8*I*a*sqrt(-2*I*e)*weierstrassPInverse(4, 0, cos (d*x + c) - I*sin(d*x + c)))/(d*e^2)]
\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=a \left (\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \] Input:
integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))**(3/2),x)
Output:
a*(Integral((e*csc(c + d*x))**(-3/2), x) + Integral(sec(c + d*x)/(e*csc(c + d*x))**(3/2), x))
\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((a*sec(d*x + c) + a)/(e*csc(d*x + c))^(3/2), x)
\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate((a*sec(d*x + c) + a)/(e*csc(d*x + c))^(3/2), x)
Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int((a + a/cos(c + d*x))/(e/sin(c + d*x))^(3/2),x)
Output:
int((a + a/cos(c + d*x))/(e/sin(c + d*x))^(3/2), x)
\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\frac {\sqrt {e}\, a \left (\int \frac {\sqrt {\csc \left (d x +c \right )}}{\csc \left (d x +c \right )^{2}}d x +\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )}{\csc \left (d x +c \right )^{2}}d x \right )}{e^{2}} \] Input:
int((a+a*sec(d*x+c))/(e*csc(d*x+c))^(3/2),x)
Output:
(sqrt(e)*a*(int(sqrt(csc(c + d*x))/csc(c + d*x)**2,x) + int((sqrt(csc(c + d*x))*sec(c + d*x))/csc(c + d*x)**2,x)))/e**2